<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd">
<html>
<head>
<title></title>
<!--Generated on Tue Jun  3 20:03:15 2014 by LaTeXML (version 0.7.999_04) http://dlmf.nist.gov/LaTeXML/.-->

<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<link rel="stylesheet" href="../../../../../../../LaTeXML.css" type="text/css">
<link rel="stylesheet" href="../../../../../../../ltx-article.css" type="text/css">
<link rel="stylesheet" href="../../../../../../../customRules.css" type="text/css">
</head>
<body>
<div class="ltx_page_main">
<div class="ltx_page_content">
<div class="ltx_document">
<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Definition of the fitting region</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">Given the approximate position of a molecule <img id="Sx1.p1.m1" class="ltx_Math" style="vertical-align:-7px" src="mi/mi100.png" width="61" height="22" alt="\left(\tilde{x}_{p},\tilde{y}_{p}\right)">
and a user-specified fitting radius <img id="Sx1.p1.m2" class="ltx_Math" style="vertical-align:-3px" src="mi/mi105.png" width="45" height="16" alt="r&gt;0">, we define <img id="Sx1.p1.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi101.png" width="235" height="21" alt="\mathcal{D}=\left\{-r,\ldots,r\right\}\times\left\{-r,\ldots,r\right\}">
as a set of integer <img id="Sx1.p1.m4" class="ltx_Math" style="vertical-align:-6px" src="mi/mi60.png" width="45" height="21" alt="\left(x,y\right)"> coordinates, and <img id="Sx1.p1.m5" class="ltx_Math" style="vertical-align:-7px" src="mi/mi102.png" width="211" height="25" alt="\tilde{I}\left(x,y\right)=I\left(x+\tilde{x}_{p},y+\tilde{y}_{p}\right)">
as intensity values of an <img id="Sx1.p1.m6" class="ltx_Math" style="vertical-align:-3px" src="mi/mi104.png" width="37" height="18" alt="l\times l"> sub-image centered at the point
<img id="Sx1.p1.m7" class="ltx_Math" style="vertical-align:-7px" src="mi/mi100.png" width="61" height="22" alt="\left(\tilde{x}_{p},\tilde{y}_{p}\right)"> of the raw input image
<img id="Sx1.p1.m8" class="ltx_Math" style="vertical-align:-2px" src="mi/mi70.png" width="13" height="16" alt="I">, where <img id="Sx1.p1.m9" class="ltx_Math" style="vertical-align:-3px" src="mi/mi103.png" width="81" height="18" alt="l=2r+1"> is the size of the sub-image. The desired sub-pixel
coordinates of the molecule are obtained as <img id="Sx1.p1.m10" class="ltx_Math" style="vertical-align:-7px" src="mi/mi97.png" width="103" height="21" alt="\hat{x}_{p}=\hat{x}_{0}+\tilde{x}_{p}">
and <img id="Sx1.p1.m11" class="ltx_Math" style="vertical-align:-7px" src="mi/mi99.png" width="99" height="21" alt="\hat{y}_{p}=\hat{y}_{0}+\tilde{y}_{p}">, where <img id="Sx1.p1.m12" class="ltx_Math" style="vertical-align:-5px" src="mi/mi96.png" width="22" height="19" alt="\hat{x}_{0}">
and <img id="Sx1.p1.m13" class="ltx_Math" style="vertical-align:-5px" src="mi/mi98.png" width="21" height="19" alt="\hat{y}_{0}"> define the sub-pixel refinements of the coordinates
obtained by the methods described below.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="Estimators.html" title="" class="ltx_ref">Sub-pixel 2D localization of molecules</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="CenterOfMassEstimatorUI.html" title="" class="ltx_ref">Centroid of local neighborhood</a></p>
</div>
</li>
<li id="I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i3.p1" class="ltx_para">
<p class="ltx_p"><a href="RadialSymmetryEstimatorUI.html" title="" class="ltx_ref">Radial symmetry</a></p>
</div>
</li>
<li id="I1.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i4.p1" class="ltx_para">
<p class="ltx_p"><a href="PSF.html" title="" class="ltx_ref">Point-spread function (PSF)</a></p>
</div>
</li>
<li id="I1.i5" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i5.p1" class="ltx_para">
<p class="ltx_p"><a href="Fitting.html" title="" class="ltx_ref">Fitting point-spread function models</a></p>
</div>
</li>
<li id="I1.i6" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i6.p1" class="ltx_para">
<p class="ltx_p"><a href="CrowdedField.html" title="" class="ltx_ref">Multiple-emitter fitting analysis</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="ltx_page_footer">
<div class="ltx_page_logo">Generated  on Tue Jun  3 20:03:15 2014 by <a href="http://dlmf.nist.gov/LaTeXML/">LaTeXML <img src="" alt="[LOGO]"></a>
</div>
</div>
</div>
</body>
</html>
